Top

Meccanica

Published in:

11-10-2020 | Original paper

Approximation of the Stokes eigenvalue problem on triangular domains using a stabilized finite element method

Author: Önder Türk

Published in: Meccanica | Issue 10/2020

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In this paper, we consider a stabilized finite element method for the approximation of the Stokes eigenvalue problem on triangular domains. The method depends on orthogonal subscales that has proved to be an appropriate means for approximating eigenvalue problems in the framework of residual based approaches. We consider several isosceles triangular domains with various apex angles to investigate the characteristics of the eigensolutions in regards to the variation of the domain properties. This study presents the first finite element approximation to the solutions of the Stokes eigenvalue problem on triangular domains, to the best of our knowledge. We provide plots of several velocity and pressure fields corresponding to the fundamental eigenmodes to analyze the flow characteristics in detail. Furthermore, we consider the problem on triangular domains including a crack, and investigate the influence of the length of the slit on the fundamental mode to some extent. The results reveal the correlation between the domain properties and the eigenpairs, and the fact that there are various critical lengths of the slit where the eigenspace is notably affected.

previous article Effect of gravity modulation on linear, weakly-nonlinear and local-nonlinear stability analyses of stationary double-diffusive convection in a dielectric liquid

next article Modified couple stress theory for three-dimensional elasticity in curvilinear coordinate system: application to micro torus panels

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Foias C, Manley O, Rosa R, Temam R (2001) Navier-Stokes equations and turbulence. Encyclopedia of mathematics and its applications. Cambridge University Press, CambridgeCrossRef

Nerli A, Camarri S (2006) Stokes eigenfunctions and galerkin projection of the disturbance equations in plane poiseuille flow: a systematic analytical approach. Meccanica 41(6):671–680MathSciNetCrossRef

Schneider K, Farge M (2008) Final states of decaying 2D turbulence in bounded domains: Influence of the geometry. Physica D: Nonlinear Phenomena 237(14–17):2228–2233MathSciNetCrossRef

Labrosse G, Leriche E, Lallemand P (2014) Stokes eigenmodes in cubic domain: their symmetry properties. Theor Comput Fluid Dyn 28(3):335–356CrossRef

Leriche E, Labrosse G (2000) High-order direct Stokes solvers with or without temporal splitting: numerical investigations of their comparative properties. SIAM J Sci Comput 22(4):1386–1410MathSciNetCrossRef

Leriche E, Labrosse G (2011) Are there localized eddies in the trihedral corners of the Stokes eigenmodes in cubical cavity? Comput Fluids 43(1):98–101CrossRef

Gedicke J, Khan A (2018) Arnold-Winther mixed finite elements for Stokes eigenvalue problems. SIAM J Sci Comput 40(5):A3449–A3469MathSciNetCrossRef

Han J, Zhang Z, Yang Y (2015) A new adaptive mixed finite element method based on residual type a posterior error estimates for the Stokes eigenvalue problem. Numer Methods Partial Differ Equ 31(1):31–53MathSciNetCrossRef

Türk Ö, Boffi D, Codina R (2016) A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems. Comput Methods Appl Mech Eng 310:886–905MathSciNetCrossRef

10.

John V, Kaiser K, Novo J (2016) Finite element methods for the incompressible Stokes equations with variable viscosity. ZAMM J Appl Math Mech 96(2):205–216MathSciNetCrossRef

11.

Du S, Duan H (2018) Analysis of a stabilized finite element method for Stokes equations of velocity boundary condition and of pressure boundary condition. J Comput Appl Math 337:290–318MathSciNetCrossRef

12.

Pearson JW, Pestana J, Silvester DJ (2018) Refined saddle-point preconditioners for discretized Stokes problems. Numer Math 138(2):331–363MathSciNetCrossRef

13.

Santo ND, Deparis S, Manzoni A, Quarteroni A (2018) Multi space reduced basis preconditioners for parametrized Stokes equations. Comput Math Appl 77:1583–1604MathSciNetCrossRef

14.

Cioncolini A, Boffi D (2019) The mini mixed finite element for the Stokes problem: an experimental investigation. Comput Math Appl 77(9):2432–2446MathSciNetCrossRef

15.

Prato Torres R, Domínguez C, Díaz S (2019) An adaptive finite element method for a time-dependent Stokes problem. Numer Methods Partial Differ Equ 35(1):325–348MathSciNetCrossRef

16.

Bustamante C, Power H, Florez W (2013) A global meshless collocation particular solution method for solving the two-dimensional Navier-Stokes system of equations. Comput Math Appl 65(12):1939–1955MathSciNetCrossRef

17.

Li M, Tang T (1996) Steady viscous flow in a triangular cavity by efficient numerical techniques. Comput Math Appl 31(10):55–65MathSciNetCrossRef

18.

Chen L, Labrosse G, Lallemand P, Luo LS (2016) Spectrally accurate Stokes eigen-modes on isosceles triangles. Comput Fluids 132:1–9MathSciNetCrossRef

19.

Cao J, Wang Z, Chen L (2017) A triangular spectral method for the Stokes eigenvalue problem by the stream function formulation. Numer Methods Partial Differ Equ 34(3):825–837MathSciNetCrossRef

20.

Yolcu SY, Yolcu T (2012) Multidimensional lower bounds for the eigenvalues of Stokes and Dirichlet Laplacian operators. J Math Phys 53(4):043508MathSciNetCrossRef

21.

Babuška I, Osborn J (1991) Eigenvalue problems. In: Finite element methods (part 1), Handbook of numerical analysis, vol 2. Elsevier, Amsterdam, pp 641–787

22.

Codina R (2000) Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput Methods Appl Mech Eng 190:1579–1599MathSciNetCrossRef

23.

Codina R (2008) Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales. Appl Numer Math 58:264–283MathSciNetCrossRef

24.

Codina R, Blasco J (1997) A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput Methods Appl Mech Eng 143:373–391MathSciNetCrossRef

Title: Approximation of the Stokes eigenvalue problem on triangular domains using a stabilized finite element method
Author: Önder Türk
Publication date: 11-10-2020
Publisher: Springer Netherlands
Published in: Meccanica / Issue 10/2020
Print ISSN: 0025-6455
Electronic ISSN: 1572-9648
DOI: https://doi.org/10.1007/s11012-020-01243-w

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Other articles of this Issue 10/2020

Variable cant angle winglets for improvement of aircraft flight performance

An analysis method of the vortex-induced vibrations of a tethered sphere

In memory of Professor Raffaele Casciaro

Anisotropic stress state and small strain stiffness in granular materials: RC experiments and DEM simulations

Numerical analysis of secondary droplets characteristics due to drop impacting on 3D cylinders considering dynamic contact angle

On magnetophoretic separation of blood cells using Halbach array of magnets

Premium Partners